Hi russkefan! Thanks for the question! In short, yes, you got the rules right for the dragon.
Persistent enemies like the dragon will attack once during combat on the turn they are rolled. Their health does not regenerate unless otherwise stated, so be sure to track any damage done to them.
Then, on this and each of the following turn's persistent phases, the dragon will attack once. In addition to this attack, the dragon will (after starting in the deepest room in your mine) climb the ladder each turn. Any room it is in, or deeper than the dragon, no longer has the positive effect for your mine.
I hope this helps. Thanks for checking out the game, I am glad you enjoyed it!